Properties

Label 5.88510464.12t183.b.a
Dimension $5$
Group $S_6$
Conductor $88510464$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(88510464\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 7^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.3687936.1
Galois orbit size: $1$
Smallest permutation container: 12T183
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.2.3687936.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 4x^{4} - 2x^{3} + x^{2} - 2x - 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: \( x^{2} + 192x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 167 a + 52 + \left(186 a + 78\right)\cdot 193 + \left(190 a + 25\right)\cdot 193^{2} + \left(157 a + 90\right)\cdot 193^{3} + \left(112 a + 99\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 49 + 23\cdot 193 + 35\cdot 193^{2} + 151\cdot 193^{3} + 93\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 26 a + 26 + \left(6 a + 98\right)\cdot 193 + \left(2 a + 29\right)\cdot 193^{2} + \left(35 a + 57\right)\cdot 193^{3} + \left(80 a + 54\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 62 + 104\cdot 193 + 147\cdot 193^{2} + 13\cdot 193^{3} + 32\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 134 a + 128 + \left(57 a + 175\right)\cdot 193 + \left(141 a + 128\right)\cdot 193^{2} + \left(69 a + 72\right)\cdot 193^{3} + \left(149 a + 13\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 59 a + 69 + \left(135 a + 99\right)\cdot 193 + \left(51 a + 19\right)\cdot 193^{2} + \left(123 a + 1\right)\cdot 193^{3} + \left(43 a + 93\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-3$
$15$$2$$(1,2)$$1$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$-1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$0$
$120$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.